我正在尝试使几个过程自动化,例如使用Sympy从Lagrangian生成ODE,并使用Numpy和Scipy对它们进行数值积分。最后的完整代码。作为生成ODE的结果,solve()
我得到了一个包含Sympy表达式的字典,如下所示:
{Derivative(lambda1(t), t): (y(t) + 1)/(x(t)*y(t)),
Derivative(z(t), t): x(t),
Derivative(x(t), t): y(t)*z(t),
Derivative(y(t), t): -x(t)*z(t)
}
然后我想将微分方程组与odeint()
Scipy集成在一起。为此,我需要lambdify
从a内的字典中提取表达式(例如)def Field(Q,t):
,以引入as odeint(Field,Q_0,t_array)
。这是我遇到困难的地方:
我第一次尝试
def Equ2(nQ,t,Q,Field):
x1,y1,z1,lamb1 = nQ
dQ =[]
for f in Q:
dQ.append(lambdify(Q, Field[f.diff(t)],'numpy' )(x1,y1,z1,lamb1))
return dQ[0:len(nQ)]
但它不能去odeint()
,因为它需要的字段2点的参数,我试图把它传递可选arga=()
的odeint()
,给我一个(长)错误:
ValueError Traceback (most recent call last)
<ipython-input-20-63f086b8a252> in Equ2(nQ, t, Q, Field)
20 dQ =[]
21 for f in Q:
---> 22 dQ.append(lambdify(Q, Field[f.diff(t)],'numpy' )(x1,y1,z1,lamb1))
23 return dQ[0:len(Q)-1]
[...]
ValueError:
Can't calculate 1st derivative wrt 14.0430379424125.
所以我尝试了基本相同的方法,但是没有循环,
def Equ1(nQ,t):
x1,y1,z1,lamb1 = nQ
dx = lambdify((x,y,z,lam[0]), field[x.diff(t)],'numpy' )(x1,y1,z1,lamb1)
dy = lambdify((x,y,z,lam[0]), field[y.diff(t)],'numpy' )(x1,y1,z1,lamb1)
dz = lambdify((x,y,z,lam[0]), field[z.diff(t)],'numpy' )(x1,y1,z1,lamb1)
dlam = lambdify((x,y,z,lam[0]), field[lam[0].diff(t)],'numpy' )(x1,y1,z1,lamb1)
return [dx,dy,dz]
并有(我认为)相同的问题:
ValueError Traceback (most recent call last)
<ipython-input-20-63f086b8a252> in Equ1(nQ, t)
9 def Equ1(nQ,t):
10 x1,y1,z1,lamb1 = nQ
---> 11 dx = lambdify((x,y,z,lam[0]), field[x.diff(t)],'numpy' )(x1,y1,z1,lamb1)
12 dy = lambdify((x,y,z,lam[0]), field[y.diff(t)],'numpy' )(x1,y1,z1,lamb1)
13 dz = lambdify((x,y,z,lam[0]), field[z.diff(t)],'numpy' )(x1,y1,z1,lamb1)
[...]
ValueError:
Can't calculate 1st derivative wrt 17.6326726993661.
如果我简单地尝试:
def Equ0(nQ,t):
x,y,z,lamb = nQ
dx = y*z
dy = -x*z
dz = x
dlam = (y+1.)/(x*y)
return [dx,dy,dz]
集成工作正常。同样,如果我EquX()
使用带有类似参数的函数调用odeint()
它们,它们将在内部正常工作。
完整代码
from sympy import *
from sympy.physics.mechanics import dynamicsymbols
from numpy import linspace, sin, cos
from scipy.integrate import odeint
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
z = Function('z')(t)
lam = dynamicsymbols('lambda1:{0}'.format(5))
f = x.diff(t)- y*z
eq = Matrix([x.diff(t) - lam[0].diff(t)*y*x*z+z,
y.diff(t) +x*z,
z.diff(t)-x
])
field = solve(list(eq)+[f],[x.diff(t),y.diff(t),z.diff(t),lam[0].diff(t)])
def Equ0(nQ,t):
x,y,z,lamb = nQ
dx = y*z
dy = -x*z
dz = x
dlam = (y+1.)/(x*y)
return [dx,dy,dz]
def Equ1(nQ,t):
x1,y1,z1,lamb1 = nQ
dx = lambdify((x,y,z,lam[0]), field[x.diff(t)],'numpy' )(x1,y1,z1,lamb1)
dy = lambdify((x,y,z,lam[0]), field[y.diff(t)],'numpy' )(x1,y1,z1,lamb1)
dz = lambdify((x,y,z,lam[0]), field[z.diff(t)],'numpy' )(x1,y1,z1,lamb1)
dlam = lambdify((x,y,z,lam[0]), field[lam[0].diff(t)],'numpy' )(x1,y1,z1,lamb1)
return [dx,dy,dz]
def Equ2(nQ,t,Q,Field):
x1,y1,z1,lamb1 = nQ
dQ =[]
for f in Q:
dQ.append(lambdify(Q, Field[f.diff(t)],'numpy' )(x1,y1,z1,lamb1))
return dQ[0:len(Q)-1]
q = [x,y,z,lam[0]]
nq = [1,2,3,4]
time=linspace(0,10,10)
### This line works just fine:
print Equ0(nq,t), Equ1(nq,t), Equ2(nq,t,q,field) #They give the same output
sol0 = odeint(Equ0,nq,time)
sol1 = odeint(Equ1,nq,time) #Errors here
sol2 = odeint(Equ2,nq,time,args=(q,field)) #And here
最后是完整错误:
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-20-63f086b8a252> in Equ1(nQ, t)
9 def Equ1(nQ,t):
10 x1,y1,z1,lamb1 = nQ
---> 11 dx = lambdify((x,y,z,lam[0]), field[x.diff(t)],'numpy' )(x1,y1,z1,lamb1)
12 dy = lambdify((x,y,z,lam[0]), field[y.diff(t)],'numpy' )(x1,y1,z1,lamb1)
13 dz = lambdify((x,y,z,lam[0]), field[z.diff(t)],'numpy' )(x1,y1,z1,lamb1)
/usr/local/lib/python2.7/dist-packages/sympy/core/expr.pyc in diff(self, *symbols, **assumptions)
2864 new_symbols = list(map(sympify, symbols)) # e.g. x, 2, y, z---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-20-63f086b8a252> in Equ1(nQ, t)
9 def Equ1(nQ,t):
10 x1,y1,z1,lamb1 = nQ
---> 11 dx = lambdify((x,y,z,lam[0]), field[x.diff(t)],'numpy' )(x1,y1,z1,lamb1)
12 dy = lambdify((x,y,z,lam[0]), field[y.diff(t)],'numpy' )(x1,y1,z1,lamb1)
13 dz = lambdify((x,y,z,lam[0]), field[z.diff(t)],'numpy' )(x1,y1,z1,lamb1)
/usr/local/lib/python2.7/dist-packages/sympy/core/expr.pyc in diff(self, *symbols, **assumptions)
2864 new_symbols = list(map(sympify, symbols)) # e.g. x, 2, y, z
2865 assumptions.setdefault("evaluate", True)
---> 2866 return Derivative(self, *new_symbols, **assumptions)
2867
2868 ###########################################################################
/usr/local/lib/python2.7/dist-packages/sympy/core/function.pyc in __new__(cls, expr, *variables, **assumptions)
1068 ordinal = 'st' if last_digit == 1 else 'nd' if last_digit == 2 else 'rd' if last_digit == 3 else 'th'
1069 raise ValueError(filldedent('''
---> 1070 Can\'t calculate %s%s derivative wrt %s.''' % (count, ordinal, v)))
1071
1072 if all_zero and not count == 0:
ValueError:
Can't calculate 1st derivative wrt 0.0.
2865 assumptions.setdefault("evaluate", True)
---> 2866 return Derivative(self, *new_symbols, **assumptions)
2867
2868 ###########################################################################
/usr/local/lib/python2.7/dist-packages/sympy/core/function.pyc in __new__(cls, expr, *variables, **assumptions)
1068 ordinal = 'st' if last_digit == 1 else 'nd' if last_digit == 2 else 'rd' if last_digit == 3 else 'th'
1069 raise ValueError(filldedent('''
---> 1070 Can\'t calculate %s%s derivative wrt %s.''' % (count, ordinal, v)))
1071
1072 if all_zero and not count == 0:
ValueError:
Can't calculate 1st derivative wrt 0.0.
TL; DR某些导出错误出现在odeint()内部,我无法使用定制函数在odeint()外部重现。
如果要t
用作符号,则应避免t
在函数声明中声明为浮点数。尝试用t
其他名称替换浮点,s
或者tt
或者...
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